Homogenization of corrugated core sandwiches panels

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Introduction
Sandwich structural panels are widely used in many industrial areas, and the purpose of the present paper is to investigate their applications in shipbuilding, as an alternative to stiffened panels.
To this aim, a comparative study of different panels will be made with respect to a reference stiffened panel similar to those traditionally used in shipbuilding. The analysis will be limited to their ratios stiffness over mass.
The sandwich panels considered here are corrugated core sandwich panels, the core and the facings being made of steel. Four different core shapes were selected for this study, owing to their easy manufacturing by laser welding. The unidirectional corrugated core is: (a) straight, (b) of hat type, (c) triangular or (d) curvilinear. The stiffened panel (e) used as a reference in the study is also presented in Fig. 1.
Several approaches to the modeling of these structural elements are discussed in the literature, see e.g. the review article [1].
For instance, a three-dimensional (3D) finite element model can be used, where the actual geometry of the core is represented. However, such a model is bound to have a high computational cost, and is inadequate when one is interested in the overall response of the structure. At the scale of a ship, these panels present indeed an overall plate-like behavior, and the problem is then to define the effective plate stiffnesses.
Several authors proposed to use an equivalent continuum instead of the core, and then combine it with a 3D model [2] or a bidimensional plate model [3] (in both references, a honeycomb core is considered. For a variety of cellular core geometries, see the recent review article [4]). In the plate model, the sandwich is then modeled as several layers, and some approximations are made in the direction of the thickness. This type of approach was used among other things for the corrugated cores that are of interest here. Then, from a strength-ofmaterials type of theory, analytical expressions for the characteristics of the equivalent continuum are given in [5,6]. Different geometries of corrugation are studied in the latter reference.
Another approach consists in using the homogenization theory, taking benefit from the periodicity exhibited by the structure in its plane. Consequently, the basic cell is made of the core and the facings, i.e. the core is not considered independently. At this stage, it is noteworthy that the panels studied here are thin 3D structures with periodic microstructure in direction 1, and invariant in direction 2 of plane (x 1 , x 2 ). These panels are therefore a special case of panels with periodic components in both directions of the plane. Such structures are characterized by the existence of two small parameters noted e and e: e is the ratio between panel thickness t and a characteristic dimension L of the panel in its plane, and e is the ratio between period length X 1 or X 2 and L (see Fig. 2).
The purpose of homogenization is then to substitute the thin initial heterogeneous structure with an equivalent homogeneous plate. This leads to making the two small parameters of the problem tend to zero. This approach was studied theoretically by Caillerie [7], where it was shown that depending on the order in which the two small parameters tend to zero, namely, one after the other or both simultaneously, varied results can be obtained for the effective stiffnesses. Therefore, several homogenization methods are available, with their own domain of validity. Our objective here is not to discuss these different methods, but its interesting to note that replacing the core by an equivalent medium amounts to make e ! 0 first, and then e ! 0 using an appropriate sandwich plate theory. The interested reader is referred to the book of Lewinski and Telega [8], where the methods are widely presented, and to [9,10]. In [8], the different methods are applied to the bending of a symmetric stiffened panel, and in [9 11] to the membrane and bending behaviors of plates with honeycomb core and of corrugated plates with straight unidirectional stiffeners.
In practice, the choice of a homogenization method is dictated by the value of ratio e=e. The panels studied here have values of e and e that are on the same order of magnitude. From a classical study of a heterogeneous plate made of a large number of basic cells, one can identify the equivalent characteristics of the plate. Then, by comparing these characteristics and those given by different homogenization methods, it appears that the most accurate results are obtained when both small parameters simultaneously become vanishingly small [10]. This is in the same line as the conclusion drawn in [8]. The latter method will therefore be used here.
This method was initially presented in [7] and is exposed in detail in [8]. A similar approach was proposed in [12] to study the bending of symmetric plates with a variable thickness. In concrete terms, the homogenization method involves basic cell problems posed on the 3D period of the panel, i.e. including both the core and the facings. Therefore, as noted in [11], this method is more appropriate than the homogenization of the core into an equivalent medium separately (see [4] for cellular sandwich cores), even if the latter may incorporate core face sheet constraints [13].
Solving the basic cell problems provides the plate effective properties according to a Kirchhoff Love model. In the literature, approximated analytical solutions to the cell problems were presented in [8] for unidirectional periodic plates, and in [14 16] for various applications (stiffened plates, sandwich plates with honeycomb core and fiber reinforced plates).
The cell problems can also be solved numerically using a finite element model of the basic cell. One can find such results in [12] for the bending of symmetric stiffened plates, in [17] for unidirectional periodic plates and in [9 11] for plates with straight unidirectional stiffeners or honeycomb core. This method of resolution presents the advantage that it can be applied to any type of period and this approach was adopted here.
In the present work, the homogenization theory, based on the asymptotic expansion method, will briefly be described in Section 2. The basic cell problems and their numerical implementation will be presented shortly after. Then, in Section 3, the homogenization method will be applied to the different panels shown in Fig. 1. By comparing the response of the equivalent homogeneous plates with that given by detailed finite element models of sandwich panels, the accuracy of the method will be   demonstrated in Section 4 for in-plane and pure bending types of loading. In Section 5, the effective properties of a fiber-reinforced plastic (FRP) honeycomb sandwich panel are computed and compared to those given in [18], in order to show the applicability of the homogenization method presented in this paper to such structures.
The main limitation of the homogenization method used here is that, at the leading order, it leads to an equivalent Kirchhoff Love homogeneous plate. For the structures studied in the present paper, it is likely that such a model will not be totally satisfactory. This problem will be dealt with in Section 6, where a simple method will be proposed to obtain the panel effective transverse shear stiffness, and finally get the properties of an equivalent Reissner Mindlin homogeneous plate.
The summation convention on repeated indices will be used throughout the paper. The Latin indices range from 1 to 3, whereas the Greek indices range from 1 to 2. In addition, dots and semi-colons will respectively note the scalar and double products of tensors, e.g.: ðr Á nÞ i ¼ r ij n j and ða : eÞ ij ¼ a ijkl e kl .

Main results
Following [7,8,10], the main steps of the homogenization method are exposed in this section.
The starting point is the formulation of the 3D problem of a thin structure with in-plane periodic components (the general case of a periodic structure in both directions of its plane is studied here). As mentioned in introduction, this problem involves two small parameters: e, which corresponds to the slenderness of the structure, and e, associated to the rapid periodic variation of the material and geometrical properties in the in-plane directions. These two small parameters are of the same order of magnitude and are assumed to be equal. Thus, one can use the asymptotic expansion method with one small parameter for periodic plates.
Firstly, two scales are introduced: a microscopic one, which is the scale of the heterogeneities and of the thickness, and a macroscopic scale on which the size of the basic cell is very small. The corresponding variables are respectively: y i ¼ x i =e and ðz 1 ; z 2 Þ ¼ ðx 1 ; x 2 Þ, where ðxÞ ¼ ðx 1 ; x 2 ; x 3 Þ is the initial 3D variable. The operators of the 3D elasticity problem are then expressed as functions of these new variables. Next, the solution of this problem is searched under the form: where functions u i ðz 1 ; z 2 ; y 1 ; y 2 ; y 3 Þ are Y 1 and Y 2 -periodic in variable y, Y a ¼ X a =e being a length of the period or basic cell Y at the microscopic scale, see Fig. 2. It will be considered that the first term u 0 ðz 1 ; z 2 Þ of the expansion in Eq. (1) has only a component in direction 3, which amounts to assuming that the plate bending is prominent, the terms relative to the in-plane displacement being of the 1th order in e, that is u 0 a ¼ 0. It then turns out that the 3D elasticity problem splits in a sequence of 2D microscopic problems, posed on the basic cell, and 2D macroscopic problems providing the overall plate response.
The main results of the method are now recalled, for more details, see [7,8,10]: • The solution of the leading order ()1th order) microscopic problem is (the solution is unique up to an additive constantû u 1 , which corresponds to a rigid body translation): where d ia is the Kronecker symbol, and whereŷ y 3 ¼ y 3 À y y 3 with y y 3 ¼ ð1=jY Ã jÞ R Y Ã y 3 dy 1 dy 2 dy 3 , and jY Ã j represents the volume of the solid part Y Ã of Y. Then, y y 3 is the distance in direction 3 from the mid-plane of the panel, such that y 3 ¼ y y 3 orŷ y 3 ¼ 0. In the following, the overall plate behavior will be formulated with respect to the mid-plane.
• The leading order macroscopic problem generalizes the Kirchhoff Love theory, the transverse and inplane displacements being u 3 0 andû u 1 a . The corresponding homogenized (or effective) constitutive relations are obtained from the solution of the 0th order microscopic problem, which will be exposed in Section 2.2.

The basic cell problems
Let us introduce membrane macrodeformations E and curvature macrodeformations K defined as: where e z denotes the strain operator corresponding to the macroscopic variable z. The 0th order microscopic problems are posed on the basic cell (see Fig. 2). The upper and lower facings are respectively noted oY þ and oY À , while oY 1 stands for the internal surfaces, which are not loaded. Denoting a the elastic moduli tensor, div y and e y the divergence and strain operators with respect to the microscopic variable y, the basic cell problems consist in finding the fields u per , e, r 1 such that: div y r 1 ¼ 0 r 1 ¼ aðyÞ : e e ab ¼ E ab þŷ y 3 K ab þ e y ab ðu per Þ e i3 ¼ e y i3 ðu per Þ r 1 Á n ¼ 0 on oY AE [ oY 1 u per y 1 y 2 per and r 1 Á n anti-per where ÔperÕ means Y 1 and Y 2 -periodic in variable y, and anti-per means that r 1 Á n are opposite on opposite sides of oY . The data of problem (4) are the macrodeformations E and K, and due to linearity, its solution is, up to a constant for the displacement: The overall plate behavior is defined from the macroscopic in-plane stress resultants and stress couples associated with r 1 . The latter are obtained through integration over the thickness and calculated using the following averaging method in period mid-planeŷ y 3 ¼ 0: The homogenized constitutive equation can then be put in the form: with the matrices ½A, ½B, ½D easily obtained from the tensors F and G introduced in Eq. (5), and the expressions given in Eq. (6). Obtaining functions v E ab and v K ab requires to solve 6 cell problems similar to problem (4), where one successively considers that the only non-zero data is a component of the membrane macrodeformation E or curvature K.

Numerical solving of homogenization problems
The basic cell problems (4) have the same characteristics as those dealt with for the homogenization of the elastic behavior of periodic composite materials (macroscopic deformation, periodicity conditions), and for which different finite element solution approaches are available in the literature [19 23].
One can mainly distinguish 2 solution techniques for the cell problems, both of them using the finite element method. The first technique consists in taking macrodeformations as problem input data and in working with stress field s 1 ¼ aðyÞ : e y ðu per Þ. The macrodeformations amount to body forces and surface loads on the edges, and the calculation of the global force vector is therefore rather tedious.
As a result, we chose a second technique, where one considers a stress field r 1 , and whose starting point is the following variational formulation of problem (4): Find u per y 1 y 2 per and E; K such that 8ũ uy 1 y 2 per and e E E; This technique was initially described in [20] for composite materials, and was used in [9,10] for plate problems. The discretized field at the finite element level is u per , but macroscopic deformations E and K are considered as additional degrees of freedom. These macroscopic degrees of freedom are associated with a virtual node connected to all the elements of the mesh. The matrix relation between the deformations and the degrees of freedom of the elements is then modified to satisfy Eq. (4) 3 and (4) 4 as shown below (let fu per e g denote the degrees of freedom associated with u per for the element considered): fu per e g fEg fKg where ½B is the element usual matrix, and fEg t ¼ fE 11 ; E 22 ; 2E 12 g, fKg t ¼ fK 11 ; K 22 ; 2K 12 g.
At this stage, let us recall that the origin ofŷ y 3 corresponds to the reference plane with respect to which the homogenized characteristics are obtained.
On the other hand, the periodicity of u per is taken into account by eliminating the redundant degrees of freedom, see [23] for example.
Finally, Eq. (9) yields the following set of matrix equations, where fN 1 g t ¼ fN 11 ; N 22 ; N 12 g, and fM 1 g t ¼ fM 11  where ½S is the stiffness matrix of the structure calculated from Eq. (9) and from the local constitutive equation.
As a consequence, it appears that the dual variables to macroscopic deformation fEg and fKg are respectively jY jfN 1 g and jY jfM 1 g, wherefrom a simple way to obtain the homogenized constitutive equation. Indeed, noting this equation in the form given in Eq. (7) and by successively imposing a component of ffEg t ; fKg t g in Eq. (10) to be equal to unity and the others to be zero, displacements v E ab and v K ab are the solution u per . The calculation of the reaction forces associated with the boundary conditions imposed on the macroscopic degrees of freedom makes it possible to build the stiffness matrix in Eq. (7) column by column. All these different steps were implemented in a finite element program. In practice, quadratic 3D solid finite elements are taken to mesh the cell in order to model the bending phenomena properly. In addition, the uniqueness of the solution to Problem (4) is ensured by zeroing u per at one node of the cell.
Because of the one-directional periodicity of the stiffeners, the homogenization problems are bidimensional, in plane (x 1 ; x 3 ). However, the finite element program was designed for more general applications (periodic panels in two directions, see Fig. 2) and the modifications mentioned above on the element matrices have only been made on the 3D solid elements. The model used was therefore fully 3D, with only one element in the direction x 2 . In the following, the finite element computation of the plate effective properties will be called the numerical homogenization method.
We will also use an analytical homogenization method. Since the cell problems are bidimensional, kinematical assumptions over the thickness make it possible to reduce the homogenization problem to that of a one-dimensional beam. Then, analytical expressions for the homogenized characteristics of the plate can be obtained [8]. Let us remark that the range of applications for such a method is narrow, because analytical expressions cannot be obtained for any shape of stiffeners. In particular, this method will only be used here for straight stiffeners, the other structures being studied by numerical method. The analytical expressions of the homogenized characteristics are given in Appendix A.
We are now going to apply the numerical and analytical homogenization methods described above to the different panels shown in Fig. 1.

Membrane and bending behaviors of the different structures
It is recalled that the objective of the present study is to calculate the static mechanical behavior of each structure in order to compare the stiffnesses of the alveolar geometries with that of the stiffened panel, for a given mass.
The usual panel taken as a reference for this study is a portion of stiffened panel between the main girders of the steel hull of a passengers ship. This stiffened plate is made out of 5-mm thick steel longitudinally reinforced by bulb flats HP 100 Â 6 (see Fig. 3).
The mechanical properties for steel are E ¼ 210 GPa, m ¼ 0:3.

Comparison indicators
In order to avoid the tedious comparison of all the components of the overall behavior matrices obtained for the different panels, it is useful to consider some characteristic values taken as indicators.
First, let us mention that the panel equivalent characteristics are determined with respect to the reference plane defined by y 3 ¼ y y 3 for all the panels except the stiffened one, for which the reference plane is the midplane of the carrier plate. On the other hand, following [7,8] and due to the symmetry properties of the cells, it can be shown that, for all the panels, the terms abcd representing respectively the membrane, membrane bending coupling and bending effective stiffnesses in matrices ½A, ½B, ½D are zero if index 1 or 2 appears an odd number of times, and matrix ½B is zero except for the stiffened plate.
Considering the compliance matrices ½A À1 and ½D À1 , one can classically [24] define moduli E m;f a and G m;f 12 respectively associated with in-plane and bending behaviors (membrane and bending moduli): Direction 1 is taken perpendicular to the corrugation (see Fig. 1), and thickness t ref equal to 5 mm, like the thickness of the sheet used for the reference panel.
The different panels can then be sorted by membrane moduli (E m 1 ; E m 2 ; G m 12 ) and by bending moduli (E f 1 ; E f 2 ; G f 12 ). These quantities are directly comparable to steel YoungÕs and shear moduli, and can readily be interpreted by engineers.

Introduction
The geometry of the reference panel is known (see Fig. 3), but that of the alveolar panels is to be determined. Since our objective is to compare the ratios between the equivalent stiffnesses and the mass of the different structures, our methodology will be as follows. After calculating the stiffness moduli E m;f a and G m;f 12 of the reference stiffened panel using the numerical homogenization method, the panel with straight stiffeners will be considered. Indeed, the analytical homogenization method can be implemented on this panel, making it possible to carry out a parametric study of its different geometric characteristics. Geometry for the panel with straight stiffeners will be determined such that its stiffness moduli lie within 5% of those of the reference panel. The other panel geometries will then be set in such a way that the mass per unit area of the panels be close to that of the panel with straight stiffeners. Finally, the numerical homogenization method will be applied to all the alveolar panels.

Study of the panel with straight stiffeners
The purpose here is to determine the geometry of the panel with straight stiffeners according to the procedure explained above. To this aim, a parametric pre-study is made using an analytical method of homogenization (the expressions used are given in Appendix A). The parameters of this study are total thickness t of the panel, thickness t f of the facings, thickness t c of the core and distance p between two stiffeners (see Fig. 4). By varying these geometric parameters in turn, the others being fixed, their respective influence on the panel overall membrane and bending behaviors can be assessed. The values that parameters t, t f , t c and p should be given can then readily be determined, for stiffness moduli E m;f a and G m;f 12 to be at least within 5% of those of the stiffened panel.
After computation, it appears that bending modulus E f 2 is mainly driven by panel total thickness t, and that the three other parameters, t f ; t c and p essentially affect the membrane moduli. A 60 mm thickness t is necessary to meet the bending stiffness E f 2 of the stiffened plate. As for the other parameters, t f ; t c and p, they must respectively be set to 2.4 mm, 1 mm and 45 mm to obtain the membrane moduli E m 1 and E m 2 of the stiffened panel. One finally gets a cell period whose geometric characteristics are defined in Fig. 4.

Geometric definition of the three other types of alveolar panels (hat type, triangular, curvilinear)
Geometry of the other alveolar panels was set in order for the panelsÕ mass per unit area to be close to that of the panel with straight stiffeners. Moreover, panel thickness t and facings thickness t f were taken identical for all the structures, so as to highlight the role of the core only. The cell periods defined are shown in Fig. 4.

Numerical homogenization of the alveolar panels
The geometric characteristics of the cell periods studied are presented Fig. 4.
These cell periods are studied using the numerical homogenization method. An example of the deformed shapes corresponding to the solution of the basic cell problems is shown in Fig. 5.
The results concerning the panelsÕ stiffness moduli and PoissonÕs ratios in membrane and bending behaviors are gathered in Table 1.

Results analysis
First of all, it is to be noticed that the discrepancies in the mass per unit area reach 1.2% at most. Then, comparing the different structures from the point of view of their ratios of effective stiffness over mass comes down to comparing their equivalent stiffnesses.
As opposed to the reference stiffened panel, the alveolar panels do not exhibit significant differences in their membrane stiffnesses E m 1 , E m 2 , G m 12 nor in their longitudinal bending stiffnesses E f 2 . However, their bending moduli E f 1 are much higher (in the orthogonal plane to corrugation) and so are the torsion moduli G f 12 . In fact, the pure bending behavior of alveolar panels is quasiisotropic, which means that E f 1 and E f 2 are on the same order of magnitude and that G f 12 is within a ðE f =2ð1 þ mÞÞ ratio, where m ¼ 0:3. On the contrary, the traditional stiffened plate features very different E f 1 and E f 2 moduli: the effect of the stiffeners is most significant in the corrugation direction (direction 2, see Fig. 1). The stiffened panel therefore exhibits a strongly anisotropic bending behavior.
On the other hand, the comparison of the six alveolar geometries between them shows that they have almost identical effective stiffnesses. This result reveals the prominent role played by the facings on the overall membrane and bending behaviors of an alveolar panel: as mentioned above about the straight stiffeners, the facings thickness drives the panel membrane behavior and the facings spacing drives the bending behavior. In comparison, the core shape has practically no influence on the membrane or bending stiffnesses.
Another way to pinpoint the important role played by the facings on the panel overall behavior consists in using the classical laminated plate theory to assess the equivalent behavior of a multilayered composite plate made of two facings and one intermediate layer whose YoungÕs modulus is negligible compared to that of the facings. One obtains: Then, it appears that the core has only a significant contribution in the case of loads in the corrugation direction and for in-plane shear of the panel.

Presentation of the method
In order to validate the homogenization method presented in Section 3, loading cases corresponding to simple macroscopic resultant stresses will be considered. At first, the response of the real heterogeneous panel predicted by the finite element method will be compared to the analytical solution of the same problem posed on the equivalent homogeneous plate. This procedure will make it possible to identify the equivalent characteristics of the homogeneous panel. These values will then be taken as references and will be checked against those obtained from the numerical homogenization method.
With the above objective in mind, let us consider a panel of finite dimension in one direction and of infinite dimension in the perpendicular direction, so that the problem is simplified into the one-dimensional cylindrical bending of an equivalent plate. As far as the finite element model of the real panel is concerned, the cylindrical bending conditions are imposed in the following way: • for the study in plane 13, only one finite element is taken in direction 2 since the structure is invariant in this direction, and the translation and rotation degrees of freedom of two nodes A and B facing each other in direction 2 are made equal. See Fig. 6(a) for the panel with 60°(T2) triangular stiffeners. All the following finite element analyses were run with the commercial package Samcef [25]. The meshes were made of 4-node Mindlin elements, and a perfect bonding between the facings and the core was assumed. • for the study in plane 23, one considers a one-period length in direction 1 because the structure is not invariant in this direction. The cylindrical bending conditions are written at nodes A and B located on the edges of the cell period. See Fig. 6(b).

Membrane loading
The loading and boundary conditions on the heterogeneous panel in plane 13 are described in Fig. 7(a).
The analysis is made on a 10-period model, and force F is calculated such that The response of the structure is then estimated using the finite element method, and half-sum u FE of the axial displacements of the two facings at a given position x 1 is computed for comparison with the analytical solution of the homogeneous equivalent plate problem.
The membrane and bending behaviors of the equivalent homogeneous plate being uncoupled, and matrix ½A in Eq. (7) being such that A 1112 ¼ A 2212 ¼ 0, the analytical solution is given as: As an example, the results obtained for the panel with T2 triangular stiffeners are presented. If u FE is plotted as a function of x 1 , one gets the curve shown in Fig. 8(a), and this curve can very well be approximated as a straight line. In addition, the slope of this straight line can be identified using the least-square method, which yields A 1111 ¼ 1:108 Â 10E6 N/mm.
On the other hand, according to the results about the T2 panel (see Table 1), the value obtained from homogenization is: There is obviously an excellent agreement between the value of the effective stiffness determined by the finite element analysis of the real heterogeneous panel and that calculated by the homogenization method.

Pure bending loading
In plane 13, the boundary conditions and forces shown in Fig. 7(b) impose a pure bending loading. Force F is such that M 11 ¼ À1000 N.
The results given by the finite element method are analyzed after determining w FE defined as the half-sum of the deflections of the two facings at a given position For the homogeneous plate, matrix ½D in Eq. (7) being such that D 1112 ¼ D 2212 ¼ 0, the analytical solution   is given as (where ' represents the length of the panel studied): For a 10-period-long panel with T2 stiffeners, the results for w FE are shown in Fig. 8(b). By determining coefficient D 1111 such that Eq. (15) best represents w FE in the least-square sense, one gets D 1111 ¼ 9:195 Â 10E8 NÁmm. And the value obtained from homogenization is (see Table 1): Like for membrane loading, there is an excellent agreement between the bending effective stiffness calculated on the real panel and that obtained from the homogenization method.
These results are as satisfactory as those shown above for the other membrane and bending characteristics, and this holds for all the panel types studied here. Another example of the accuracy of the homogenization method used in the present paper can be found in [9,10] where the example of a honeycomb plate made of steel is treated. In order to show the applicability of the homogenization method to composite structures, an example with FRP materials is treated in Section 5.

Homogenization of a fiber-reinforced plastic honeycomb sandwich panel
As an example of FRP structure, we consider the case of a honeycomb sandwich panel for highway decks, which was studied in [18]. This structural panel consists of a sinusoidal wave core configuration in the plane, extending vertically between face laminates. Thus, the panel exhibits in-plane periodic structure, and Fig. 9(a) presents its basic cell, the upper facing being not represented.
The core has a sinusoidal wave configuration, with an amplitude 2h (with h ¼ 25:4 mm) and p 2 ¼ 4h þ 4t c , see Fig. 9(a), one half of the core wall being modeled on the basic cell boundaries normal to direction 2.
The core and face laminates are constituted with Eglass fibers and polyester resin. The core is made with a single layer of continuous strand mat with continuous randomly oriented fibers, while the face laminates include four types of fiber layers, see [18]. However, to solve the basic cell problems, it is indeed sufficient to consider the core (with its real sinusoidal geometry) and laminates as homogeneous layers. So we use here the equivalent properties given in [18], and obtained from a micromechanics approach for each layer, combined with classical lamination theory for the facings. These material properties are listed in Tables 2 and 3,  and are taken from Tables 3 and 4 Table 2, the direction 3 is normal to the core layer, while in Table 3, the directions 1 and 2 correspond to those of Fig. 9(a)).

of [18] (in
Let us mention that given that the basic cell problems are 3D, a 3D constitutive behavior has to be defined from the data of Tables 2 and 3. So the following assumptions were made for the core properties ( Fig. 9. Basic cell geometry of the FRP honeycomb sandwich panel (a) and its finite element modeling (b). G 13 ¼ G 23 and m 13 ¼ m 23 ), while for the face laminates we took E 3 ¼ E 2 , m 13 ¼ m 23 ¼ m 12 and G 12 ¼ G 23 ¼ G 13 . One can easily verify that these assumptions have a negligible effect on effective properties since itÕs mainly the in-plane behavior of the core and the facings which is involved in the loadings of the basic cell problems. The mesh used for the numerical homogenization method is shown in Fig. 9(b).
From (11), with t ref ¼ 126:52 mm i.e. the panel total thickness, the effective moduli of the sandwich structure are computed. Table 4 shows the results obtained with the numerical 3D homogenization method, in comparison to those of [18]. In [18], in a first step, an analytical solution is used for the evaluation of equivalent core properties, and then classical lamination theory provides the sandwich panel effective properties, considering the structure as a three layers laminated system. Therefore, skin core interactions are neglected, which yields underestimate of stiffness, as noted in [11]. These effects are rigorously taken into account in the method used in this paper, through the 3D modeling used in the basic cell problems. From Table 4, it may be seen that there is not a significant difference between our approach and the results of [18]. However, our approach is more accurate because it enables to compute local distribution of stresses. Thus, once the computations are made on the equivalent homogeneous plate, these local stresses are simply obtained by a linear combination of the obtained macroscopic strains and the solution of the basic cell problems (see e.g. Fig.  10 the von-Mises stress distribution for a given unit curvature).

Introduction
Owing to what has been shown above, it can be stated that the homogenization method proves to be very accurate to provide the characteristics of an equivalent Kirchhoff Love plate. Nevertheless, it is not guaranteed that a Kirchhoff Love model is sufficient to describe the overall behavior of a sandwich panel. In order to investigate this question, let us consider a simple bending loading, still under the assumption of cylindrical bending.
The analysis is performed on a 10-period-long panel, in plane 13, when conditionũ u ¼0 0 is imposed at all the nodes in the cross-section of the origin, see Fig. 7(c). Force F is taken such that When the analysis of the heterogeneous panel is done, facings average deflection w FE is determined and compared with that given by the analytical solution to the Kirchhoff Love plate problem, i.e.: The plot shown in Fig. 11(a) represents the results obtained for a panel with T2 stiffeners. One can notice a discrepancy between the two solutions. This discrepancy   Table 3 Engineering constants for face laminates is not significant for this type of stiffeners but is important for stiffeners that are straight (see Fig. 11(b)), curvilinear, or in hat shape, although all the structures studied have almost the same equivalent membrane and bending characteristics. It appears that a Kirchhoff Love model is not completely satisfactory, and it is therefore necessary to take into account the transverse shear effects. As has been seen in the last results, these characteristics seem to be very sensitive to the core shape.
The problem is then to define an homogenization method for the transverse shear behavior.
Starting from 3D elastic problem, the asymptotic expansion method enables to recover, at the leading order, the Kirchhoff Love model. It can indeed be shown that the transverse shear forces associated with stresses r 1 13 and r 1 23 are zero. As a result, solving the cell problems described in Eq. (4) does not provide any information on the plates stiffness in transverse shear.
Refined plate theories, taking into account transverse shear effects, can be obtained using the asymptotic expansion method, including the higher-order terms of the expansion [26]. Such theories are complex, since they involved higher derivatives, and are therefore not used by engineers. This justifies attempts to derive an asymptotically correct Reissner Mindlin second order theory [27]. However, such a process isnÕt straightforward, because one has to choose the asymptotical correctness criterion. Moreover a higher-order microscopic problem needs to be solved. Consequently, this method will not be used in this paper.
Another way to proceed to obtain an equivalent Reissner Mindlin plate is to use the homogenization method proposed in [28]. Its starting point is the bidimensional equations of a Reissner Mindlin plate with periodic coefficients. As a consequence, the downside of this method is that the 3D behavior of the plate is only roughly taken into account, since it is a priori assumed that the heterogeneous 3D panel behaves like a heterogeneous Reissner Mindlin plate. The application field of this method is therefore limited to cells whose in-plane dimensions are very small compared to their thickness [8,15,16], i.e. for e ( e, whereas for the example under consideration here, e % e. It will be seen below that this method is inappropriate as far as the determination of the transverse shear stiffness is concerned.
It is then necessary to resort to another type of approach.
In the present work, we will use a numerical method to determine the corrugated core sandwich panel equivalent stiffness in transverse shear. Assuming the overall behavior being of Reissner Mindlin type, the transverse shear stiffness will be identified from the results obtained about a panel made of several cell periods. This is a difference from the method used for the homogenization of membrane and bending behaviors, where the finite element model only represented one period.
Let us mention that in the following, our study will be restricted to the corrugated sandwich panels of Fig. 1. However, the method proposed in this paper is also valid for the FRP honeycomb sandwich panel of Section 5 (in [18], its shown that a Timoshenko beam model is in good agreement with experimental data for three-point and four-point bendings).

Principle of the method
Let us re-consider the calculation made at the beginning of this section, assuming that the overall behavior in transverse shear and the membrane and bending behaviors are uncoupled, and that the associated stiffness matrix is diagonal, such that: where h 1 and h 2 denote rotations about the x 1 and x 2 axes, respectively. Then, the analytical solution to the equivalent homogeneous Reissner Mindlin plate problem is given by: One can further define from the finite element analysis of the heterogeneous panel: calculated with the value of D 1111 obtained from the homogenization method, and the accuracy of which has been established in Section 4.3. It is therefore assumed that Eq. (7) describes the membrane and bending behaviors of the equivalent Reissner Mindlin plate, when curvatures fK 11 ; K 22 ; 2K 12 g are substituted with fh 2;1 ; À h 1;2 ; h 2;2 À h 1;1 g. Then, if w Ã FE is plotted as a function of x 1 , the graphs shown in Fig. 12(a) and (b) are obtained for T2 and straight stiffeners respectively: A straight line is found in both cases, and this shows, from Eqs. (19) and (20), that the panel overall response is similar to that predicted by a Reissner Mindlin model. It also provides a simple way to identify k 1 from the slope of the straight line using the least-square method.
However, one can notice that for T2 stiffeners, the difference between w Ã FE and w FE is only about 10%. Since w Ã FE is obtained from the homogenized bending characteristics of the panel, one may want to diminish the relative importance of w KL in Eq. (20). Then, the length of the structure will be decreased so that w Ã FE is at least 20% lower than w FE at the loaded tip of the panel.
For the study in plane 13, this has led us to generally consider 5-period-long panels, except for the T1 panel, where 3 periods are taken.
In plane 23, the same procedure applies, and according to the same criterion, 5-period-long panels are considered, except for T1 and H2 panels, where 4 periods are studied. The boundary conditionũ u ¼0 0 is imposed at all the nodes in the cross-section of the origin (core and facings). One finally uses the expression: knowing that the analytical solution w Ã is given by:

Results
For all the panels studied here, straight lines very well approximate the curves w Ã FE ðx 1 Þ in plane 13 and w Ã FE ðx 2 Þ in plane 23 obtained from the method described above. In turn, the values of k 1 and k 2 can be calculated. From these values, and using reference thickness h ref ¼ 5 mm, we may define engineering constants in the form of shear moduli G 13 and G 23 such that: The values calculated from Eq. (23) are gathered in Table 5.
Let us note that if the homogenization method [28] mentioned at the beginning of Section 6 is used (see the corresponding analytical expressions given in Appendix A) one gets G 13 ¼ 0:792 GPa and G 23 ¼ 0:974 GPa for straight stiffener. These values are very far from the reference values obtained from the finite element analyses. Thus, such a homogenization method should not be applied to the type of panels considered here.
The values in Table 5 are to be compared with the shear modulus of steel G ¼ 0:808 GPa, this value being a first approximation of the stiffened panel equivalent moduli in planes 13 and 23. Indeed, it can be stated that the stiffness in transverse shear of the stiffened panel is at least that of the underlying steel plate.

Results analysis
Compared to the case of the membrane and bending equivalent characteristics, significant differences can be noted.
First of all, the value of G 13 is very different from one panel to the other, and can be up to 1000 times lower than that of the reference stiffened panel. The results in plane 23 are less scattered, and the stiffness of the sandwich panels is on the same order of magnitude as that of the stiffened panel, although lower.
As a result, the alveolar panels have a strong anisotropic behavior in transverse shear, except for panels with triangular-shaped stiffeners. It appears then that the core shape has a strong influence on the value of modulus G 13 , and that T2 triangular stiffeners at 60°give the best transverse shear stiffness for sandwich panels. Finally, the values obtained show that it is necessary to use a Reissner Mindlin plate model, even for the stiffest panel (see Fig. 11(a)).

Conclusions
In this work, we have applied a periodic homogenization method to determine the equivalent membrane and pure bending characteristics of periodic plates. This method requires a few developments in a finite element package, and is then very easy to implement. As demonstrated in Section 4, this method yields very good results. Moreover, this method can be applied to many Moduli G 13 and G 23 for a 5 mm thick equivalent homogeneous plate. engineering structures, such as composite structures, see Section 5. However, the method does not make possible the evaluation of the transverse shear stiffness of sandwich panels. Therefore, we used another approach based on calculations in cylindrical bending. Then, after verifying that the panel overall response was in agreement with a Reissner Mindlin model, its transverse shear stiffness was identified from the results of the calculations.
This study allows to compare different types of alveolar structures with traditional stiffened structures, with a view to their applications in shipbuilding. Several conclusions can be drawn from the present work.
For a given mass per unit area, alveolar panels present a better overall bending stiffness than traditional stiffened panels. On the opposite, effective membrane stiffnesses are similar for all the panels. As a result, a weight reduction is possible, especially as far as the bending characteristics are concerned.
For the overall membrane and pure bending behaviors, the two driving parameters are the thickness of the facings and their spacing, whereas stiffeners geometry has only a minor influence. However, the equivalent behavior in transverse shear is very dependent on the stiffeners geometry, and it appears that triangular stiffeners have the best characteristics.
Finally, the important role played by shear stress in the behavior of such sandwich structures was highlighted.
Thanks to the methods presented in this work, we can determine the panelsÕ effective characteristics according to a Reissner Mindlin plate. This will make it possible to model the alveolar panels as equivalent homogeneous plates, and will reduce the computation time when the finite element analysis is made on the whole ship.